Some individuals literally have a calling to do this. St. Francis, for instance, gave up everything he ever had in order to follow Christ. Many of us are called to a different life - one of marriage, Children, or even the single life. When we are called to this, we have to try and figure out how Christ's message applies to us. One way we as Catholics do penance - or offer reparation for our sins in order to help negate temporal punishment associated with them - is to give up certain things. Most famously, we abstain from meat on Fridays during the Lenten Season, and fast on certain Holy Days of Obligations. Many will go on to extend that abstention to all Fridays in the year, or give up other things.
We are definitely called to give up the pleasures associated with sin when we follow Christ. That is not an option. But, I encourage you to go even further. Think of something that you depend on, or use as a crutch, or just really enjoy. Give that up on one day a week or for a month and dedicate that saved time or money to Christ. In doing so, you will draw closer to him and lay up treasure in Heaven.